Problem: What is the slope of the line tangent to $f(x) = -2x^{2}+x+7$ at $x = -1$ ?
Solution: The slope of the tangent line is $ \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(-2(x+\Delta x)^{2}+x+\Delta x+7) - (-2x^{2}+x+7)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(-2(x^{2}+2x \Delta x+\Delta x^{2})+x+\Delta x+7) - (-2x^{2}+x+7)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{-2x^{2}-4(x \Delta x)-2\Delta x^{2}+x+\Delta x+7+2x^{2}-x-7}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{-4(x \Delta x)-2\Delta x^{2}+\Delta x}{\Delta x}$ $ = \lim_{\Delta x \to 0} -4x-2(\Delta x)+1$ $ = -4x+1$ $ = (-4)(-1)+1$ $ = 5$